Page 277 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 29: Computing Galois Groups

CONTENTS
Objectives

Introduction
29.1 Transitively Group
29.2 Separable Polynomial

29.3 Summary
29.4 Keywords

29.5 Review Questions
29.6 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss transitively and transitive group

Explain computing Galois group

Introduction

In the last unit, you have studied about Galois theory. In this unit, you will get information
related to computing the Galois groups.

29.1 Transitively Group

Definition: Let G be a group acting on a set S. We say that G acts transitively on S if for each pair
of elements x,y in S there exist an element g in G such that y = gx.
If G is a subgroup of the symmetric group S , then G is called a transitive group if it acts
n
transitively on the set { 1, 2, ... , n }.
29.2 Separable Polynomial

Proposition: Let f(x) be a separable polynomial over the field K, with roots r , ... , r in its
n
1
splitting field F. Then f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the
roots of f(x).
Lemma: Let p be a prime number, and let G be a transitive subgroup of S . Then any nontrivial
p
normal subgroup of G is also transitive.
Lemma: Let p be a prime number, and let G be a solvable, transitive subgroup of S . Then G
p
contains a cycle of length p.
Proposition: Let p be a prime number, and let G be a solvable, transitive subgroup of S . Then G
p
is a subgroup of the normalizer in S of a cyclic subgroup of order p.
p

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